with anisotropic Gibbs{Thomson law The Stefan problem

Introduction Anisotropic energies and weak formulation Main existence result Computational results

The Stefan problemwith anisotropic Gibbs–Thomson law

Harald Garcke

University of Regensburg

joint with Stefan Schaubeck (Regensburg)

August 2010

Harald Garcke The Stefan problem with anisotropic Gibbs–Thomson law

Outline

1 Introduction

2 Anisotropic energies and weak formulation

3 Main existence result

4 Computational results

Stefan problem modelsmelting and solidification

Gibbs Thomson lawaccounts for surface energy

Geometry:

Stefan problem modelsmelting and solidification

Gibbs Thomson lawaccounts for surface energy

Geometry:

Mathematical description

Energy balance

∂t(u + χ)−∆u = f weakly !

u : temperatureχ : characteristic function of liquidf : heat sources

Strong formulation:in bulk ∂tu −∆u = f heat equation

on interface Γ V = −[∂u∂ν ] Stefan condition

V : normal velocity[.] : jump across interface

Mathematical description

Energy balance

∂t(u + χ)−∆u = f weakly !

u : temperatureχ : characteristic function of liquidf : heat sources

Strong formulation:in bulk ∂tu −∆u = f heat equation

on interface Γ V = −[∂u∂ν ] Stefan condition

V : normal velocity[.] : jump across interface

Further condition on interface

Classical condition

u = 0 (= melting temperature)

Physically observed : undercooling (u < 0) in liquid andsuperheating (u > 0) for solid is possible

Reason: Surface energy effects are neglected in classical model

Modified condition

u = H (Gibbs–Thomson law)

H : mean curvature

Further condition on interface

Classical condition

u = 0 (= melting temperature)

Physically observed : undercooling (u < 0) in liquid andsuperheating (u > 0) for solid is possible

Reason: Surface energy effects are neglected in classical model

Modified condition

u = H (Gibbs–Thomson law)

H : mean curvature

Issues:

Interfaces do not stay smooth (e.g. topological changes)We need a weak formulation of u = H

u = H describes isotropic surface energyIn nature: surface energies are anisotropic (leads to crystals)

Weak formulation of u = H:

∫ΩT

(∇ · ξ − ∇χ

|∇χ|· Dξ ∇χ|∇χ|

)d |∇χ(t)|dt =

∫ΩT

∇ · (uξ)χdxdt

for all ξ ∈ C∞(ΩT ,Rn) with ξ · n = 0

Issues:

Interfaces do not stay smooth (e.g. topological changes)We need a weak formulation of u = H

u = H describes isotropic surface energyIn nature: surface energies are anisotropic (leads to crystals)

Weak formulation of u = H:

∫ΩT

(∇ · ξ − ∇χ

|∇χ|· Dξ ∇χ|∇χ|

)d |∇χ(t)|dt =

∫ΩT

∇ · (uξ)χdxdt

for all ξ ∈ C∞(ΩT ,Rn) with ξ · n = 0

Important analytical results:

Classical solutions : Chen, Reitich (1992)Radkevich (1992)Escher, Pruss, Simonett (2003)Mucha (2005)

Weak solutions : Luckhaus (1990)Roger (2004)

Idea for the construction of weak solutions (Luckhaus)Use implicit time discretization

All results are for the isotropic case

Important analytical results:

Classical solutions : Chen, Reitich (1992)Radkevich (1992)Escher, Pruss, Simonett (2003)Mucha (2005)

Weak solutions : Luckhaus (1990)Roger (2004)

Idea for the construction of weak solutions (Luckhaus)Use implicit time discretization

All results are for the isotropic case

Anisotropic energy

Interfacial energy:

F(Γ) :=

∫Γγ(ν)dHn−1 for a hypersurface γ

γ : Rd \ 0 → R+, one-homogeneous, strictly convex, smooth

First variation

δFδΓ

(Γ)(ξ) =

∫ΓHγ(ξ · ν)dHn−1

with Hγ := ∇Γ · (Dγ(ν))where Dγ is the gradient of γ

Anisotropic Gibbs–Thomson law

u = Hγ , (Gurtin, 1988)

Anisotropic energy

Interfacial energy:

F(Γ) :=

First variation

δFδΓ

(Γ)(ξ) =

u = Hγ , (Gurtin, 1988)

Anisotropic energy

Interfacial energy:

F(Γ) :=

First variation

δFδΓ

(Γ)(ξ) =

u = Hγ , (Gurtin, 1988)

Weak formulation of anisotropic Gibbs–Thomson law∫Ω

(div ξγ(ν)− ν · DξDγ(ν))d |∇χ| =

div (uξ)χdx

Question:How can we define the energy in the BV–context?

Introduce dual function !

γ0(q) = supp∈Rd\0

p · qγ(p)

q ∈ Rd \ 0

γ0 one-homogeneous, strictly convex, smooth

div (uξ)χdx

p · qγ(p)

q ∈ Rd \ 0

div (uξ)χdx

p · qγ(p)

q ∈ Rd \ 0

Important identity

F(Γ) =

∫Γγ(ν)dHn−1 = sup

∫Ωχ divϕdx | γ0(ϕ) ≤ 1

see Amar and Bellettini

Remarks

Above identity implies lower semi-continuity of F w.r.t. L1

We can show: first variation of F is given as:∫Ωdivξγ(ν)− ν · DξDγ(ν)d |∇χ|

Idea: Use sup-definition above and approximation arguments.

Important identity

F(Γ) =

∫Γγ(ν)dHn−1 = sup

∫Ωχ divϕdx | γ0(ϕ) ≤ 1

see Amar and Bellettini

Remarks

Above identity implies lower semi-continuity of F w.r.t. L1

We can show: first variation of F is given as:∫Ωdivξγ(ν)− ν · DξDγ(ν)d |∇χ|

Idea: Use sup-definition above and approximation arguments.

Visualize surface energy with the help ofFrank diagram: 1-ball of γ

F = p ∈ Rd | γ(p) ≤ 1Wulff shape: 1-ball of γ0

W = q ∈ Rd | γ0(q) ≤ 1

Isoperimetric characterization

Wulff shape W minimizes surface energy among all surfaces thatenclose the same volume

Frank diagrams and Wulff shapes for different surface energies.

Cubic anisotropy (left) and hexagonal anisotropy (right).

Visualize surface energy with the help ofFrank diagram: 1-ball of γ

F = p ∈ Rd | γ(p) ≤ 1Wulff shape: 1-ball of γ0

W = q ∈ Rd | γ0(q) ≤ 1

Isoperimetric characterization

Wulff shape W minimizes surface energy among all surfaces thatenclose the same volume

Frank diagrams and Wulff shapes for different surface energies.

Cubic anisotropy (left) and hexagonal anisotropy (right).

Main existence result

Theorem: Assumptions:

i) Ω is a bounded C 1–domain

ii) initial data u0 ∈ L∞ ∩ H1,2, χ0 ∈ BV (Ω, 0, 1),

iii) Dirichlet data uD ∈ H1,2(Ω),

iv) r.h.s f ∈ L∞(ΩT ).

Then there exists

χ ∈ L∞(ΩT , 0, 1), |∇χ(t)| ≤ C

u ∈ [uD + L2(0,T ; H1,20 (Ω))] ∩ L∞(0,T ; L2(Ω))

such that

Main existence result (continued)

∫ T0

∫Ω(u + χ)∂tϕ+

∫Ω(u0 + χ0)ϕ(0) =∫ T

∫Ω∇u · ∇ϕ−

∫ T0

∫Ω f ϕ

for all ϕ ∈ C∞0 ([0,T )× Ω), and∫ T

(div ξDγ(ν) · ν − ν · DξDγ(ν)) d |∇χ(t)|dt =

∫ΩT

div (uξ)

for all ξ ∈ C 1(ΩT ,Rd) with ξ · n = 0 on ∂Ω.

As Luckhaus use an implicite time discretization:

h > 0 time step .

Assume uh(t − h), χh(t − h) is given.Define functional

Fh,t(χ) :=∫

Ω γ(ν)d |∇χ| − 12 (u(χ))2 − u(χ)χ+

u(χ)(uh(t − h) + χh(t − h))

such that

(u + χ)− (uh(t − h) + χh(t − h))− h∆u = hf h(t)

+ b.c.Fh,t can be interpreted as negative entropy !

Fh,t has minimizer χh(t) (use direct method)Note: γ is convexDefine uh(t) as a solution of time discrete diffusion equation

Lemma (energy estimate)

ess sup0≤t≤T

(‖uh(t)‖2

∫Ωγ(νh)

∫Ω|∇uh|2 ≤ C

0‖∂−h

t (uh + χh)(t)‖2H−1,2(Ω) ≤ C

Proof: Test diffusion equation by uh − uD and use

Fh,t(χhi ) ≤ Fh,t(χh

i−1)

ess sup0≤t≤T

(‖uh(t)‖2

∫Ωγ(νh)

∫Ω|∇uh|2 ≤ C

0‖∂−h

t (uh + χh)(t)‖2H−1,2(Ω) ≤ C

i−1)

ess sup0≤t≤T

(‖uh(t)‖2

∫Ωγ(νh)

∫Ω|∇uh|2 ≤ C

0‖∂−h

t (uh + χh)(t)‖2H−1,2(Ω) ≤ C

i−1)

Use standard compactness results and compactness results of Altand Luckhaus to obtain convergent subsequence of (uh, χh)Passing to the limit in diffusion equation is standard

Minima of (χh, uh) of Fh,t fulfills

0 =∫

Ω(div ξDγ(νh(t)) · νh(t)−νh(t) · DξDγ(νh(t)))d |∇χh(t)| −

∫Ω div (uh(t)ξ)χh(t)+ (1)∫

Ω div (L0h(uh(t)− uh(t − h) + χh(t)− χh(t − h))ξ)χh(t)

for all ξ ∈ C∞(Ω,Rd), with ξ · n = 0, where νh(t) = − ∇χh(t)

|∇χh(t)|where v = L0

h(g) is the solutionof v − h∆v = −g in Ω v = 0 on ∂Ω

Goal: Pass to the limit in the discrete Gibbs–Thomson equation (1)

Minima of (χh, uh) of Fh,t fulfills

0 =∫

Ω(div ξDγ(νh(t)) · νh(t)−νh(t) · DξDγ(νh(t)))d |∇χh(t)| −

∫Ω div (uh(t)ξ)χh(t)+ (1)∫

Ω div (L0h(uh(t)− uh(t − h) + χh(t)− χh(t − h))ξ)χh(t)

for all ξ ∈ C∞(Ω,Rd), with ξ · n = 0, where νh(t) = − ∇χh(t)

|∇χh(t)|where v = L0

h(g) is the solutionof v − h∆v = −g in Ω v = 0 on ∂Ω

Goal: Pass to the limit in the discrete Gibbs–Thomson equation (1)

We observe∫Ωγ(νh)d |∇χh| →

∫Ωγ(ν)d |∇χ| as h→ 0

Uselower semicontinuity of

∫Ω γ(ν)d |∇χ|

strong convergence of uh in L2,

the fact that χh solve a minimization problem involving∫Ω γ(ν)d |∇χ|

Main difficulty: Show Dγ(νh)→ Dγ(ν)[Convergence in the Cahn–Hoffmann ξ–vector]

Since∫Ωγ(ν)|∇χ| = sup

∫Ωχ divϕdx | ϕ ∈ C 1(Ω,Rd), γ0(ϕ(x)) ≤ 1

we obtain a gε ∈ C 1

0 (Ω,Rd) with γ0(g ε) ≤ 1 such that∫Ω

(γ(ν)− gε · ν)d |∇χ| ≤ 13ε

Claim:

gε is a good approximation of the Cahn–Hoffmann vectorξ = Dγ(ν)

Technical observation

If γ is strictly convex in the sense thatthere exists a d0 > 0 such that

(D2γ0)(p)q · q ≥ d0|q|2 for all p, q ∈ Rn, |p| = 1, p · q = 0

Then we obtain (using a Lemma of Dziuk)∃C > 0 ∀ ν ∈ Sd−1, p ∈ Rd with γ0(p) ≤ 1

C |Dγ(ν)− p| ≤ γ(ν)− p · ν

Henceγ(ν)− p · ν small implies

p is a good approximation of the ξ–vector Dγ(ν)

Technical observation

If γ is strictly convex in the sense thatthere exists a d0 > 0 such that

(D2γ0)(p)q · q ≥ d0|q|2 for all p, q ∈ Rn, |p| = 1, p · q = 0

Then we obtain (using a Lemma of Dziuk)∃C > 0 ∀ ν ∈ Sd−1, p ∈ Rd with γ0(p) ≤ 1

C |Dγ(ν)− p| ≤ γ(ν)− p · ν

Henceγ(ν)− p · ν small implies

p is a good approximation of the ξ–vector Dγ(ν)

Steps: ∫Ω

(γ(ν)− gε · ν))d |∇χ| ≤ 13ε∫

Ωγ(νh)d |∇χh| →

∫Ωγ(ν)d |∇χ|

implies

gε is a good approximation of Dγ(ν) and Dγ(νh)

This can be used to show∫Ωνh · DξDγ(νh)d |∇χh| →

∫Ων · DξDγ(ν)d |∇χ|

Convergence in the other terms of the weak formulation is easier(and omitted here)We finally obtain the weak formulation∫ T

(div ξDγ(ν) · ν − ν · Dξ(ν)Dγ(ν))d |∇χ| =

∫ΩT

div (uξ)χ

with anisotropic Gibbs{Thomson law The Stefan problem

Documents

Transcript of with anisotropic Gibbs{Thomson law The Stefan problem